integer_polynomial_prime_power_hensel_iterated_exists_unique

Exact expanded first-order arithmetic statement

forall pb pc nb nc a l p k j m. ((~(p = 1) /\ forall frp_prime_left_sph_iterated_prime frp_prime_right_sph_iterated_prime. p = frp_prime_left_sph_iterated_prime * frp_prime_right_sph_iterated_prime -> frp_prime_left_sph_iterated_prime = 1 \/ frp_prime_right_sph_iterated_prime = 1)) -> ~(k = 0) -> (exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = k) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (m) = S ((S (k)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (k)) * pa_v_hpl_power_product) + (m))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = k) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))) -> (exists sph_vp_simple sph_dp_simple sph_vn_simple sph_dn_simple. ((((exists ff_u_hd_hpl_sph_simple_positive ff_v_hd_hpl_sph_simple_positive ff_d_hd_hpl_sph_simple_positive ff_e_hd_hpl_sph_simple_positive. ((((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_start. fs_h_ph_hd_hpl_sph_simple_positive_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_start. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_simple_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_terminal. fs_h_ph_hd_hpl_sph_simple_positive_body_value_terminal + S (sph_vp_simple) = S ((S (l)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_terminal. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_simple_positive) + (sph_vp_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps. (exists ph_bound_hd_hpl_sph_simple_positive_body_value_steps. ph_bound_hd_hpl_sph_simple_positive_body_value_steps + S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * pc)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient. pb = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * pc) + (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_before. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_before. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_after. fs_h_ph_hd_hpl_sph_simple_positive_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_after. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_value_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_simple_positive_body_value_steps = ff_previous_ph_hd_hpl_sph_simple_positive_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_simple_positive_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_start. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_start. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_simple_positive) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_terminal. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_terminal + S (sph_dp_simple) = S ((S (l)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_terminal. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_simple_positive) + (sph_dp_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps. (exists ph_bound_hd_hpl_sph_simple_positive_body_derivative_steps. ph_bound_hd_hpl_sph_simple_positive_body_derivative_steps + S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_positive) + (ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive) + (ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive)) /\ exists fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after. ff_d_hd_hpl_sph_simple_positive = fs_q_ph_hd_hpl_sph_simple_positive_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_positive_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_positive) + (ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_simple_positive_body_derivative_steps = ff_previous_ph_hd_hpl_sph_simple_positive_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_simple_positive_body_derivative_steps)))))))) /\ (exists ff_u_hd_hpl_sph_simple_negative ff_v_hd_hpl_sph_simple_negative ff_d_hd_hpl_sph_simple_negative ff_e_hd_hpl_sph_simple_negative. ((((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_start. fs_h_ph_hd_hpl_sph_simple_negative_body_value_start + S (0) = S ((S (0)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_start. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_start * S ((S (0)) * ff_v_hd_hpl_sph_simple_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_terminal. fs_h_ph_hd_hpl_sph_simple_negative_body_value_terminal + S (sph_vn_simple) = S ((S (l)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_terminal. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_terminal * S ((S (l)) * ff_v_hd_hpl_sph_simple_negative) + (sph_vn_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps. (exists ph_bound_hd_hpl_sph_simple_negative_body_value_steps. ph_bound_hd_hpl_sph_simple_negative_body_value_steps + S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * nc)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient. nb = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * nc) + (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_before. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_before. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_after. fs_h_ph_hd_hpl_sph_simple_negative_body_value_steps_after + S (ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_after. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_value_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_value_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps))) /\ ff_current_ph_hd_hpl_sph_simple_negative_body_value_steps = ff_previous_ph_hd_hpl_sph_simple_negative_body_value_steps * a + ff_coefficient_ph_hd_hpl_sph_simple_negative_body_value_steps)))))) /\ (((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_start. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_start + S (0) = S ((S (0)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_start. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_start * S ((S (0)) * ff_e_hd_hpl_sph_simple_negative) + (0))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_terminal. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_terminal + S (sph_dn_simple) = S ((S (l)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_terminal. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_terminal * S ((S (l)) * ff_e_hd_hpl_sph_simple_negative) + (sph_dn_simple))) /\ forall ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps. (exists ph_bound_hd_hpl_sph_simple_negative_body_derivative_steps. ph_bound_hd_hpl_sph_simple_negative_body_derivative_steps + S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps = l) -> exists ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps. ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient + S (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient. ff_u_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_coefficient * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_v_hd_hpl_sph_simple_negative) + (ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before + S (ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_before * S ((S (ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative) + (ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ((((exists fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after. fs_h_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after + S (ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps) = S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative)) /\ exists fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after. ff_d_hd_hpl_sph_simple_negative = fs_q_ph_hd_hpl_sph_simple_negative_body_derivative_steps_after * S ((S (S ff_i_ph_hd_hpl_sph_simple_negative_body_derivative_steps)) * ff_e_hd_hpl_sph_simple_negative) + (ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps))) /\ ff_current_ph_hd_hpl_sph_simple_negative_body_derivative_steps = ff_previous_ph_hd_hpl_sph_simple_negative_body_derivative_steps * a + ff_coefficient_ph_hd_hpl_sph_simple_negative_body_derivative_steps)))))))))) /\ ((exists hgcrt_mod_left_hpl_simple hgcrt_mod_right_hpl_simple. sph_vp_simple + m * hgcrt_mod_left_hpl_simple = sph_vn_simple + m * hgcrt_mod_right_hpl_simple) /\ (exists sph_inverse_simple. ((exists hpl_gap_simple. hpl_gap_simple + S (sph_inverse_simple) = (p)) /\ (exists hgcrt_mod_left_hpl_simple hgcrt_mod_right_hpl_simple. (sph_dp_simple * sph_inverse_simple) + p * hgcrt_mod_left_hpl_simple = (1 + sph_dn_simple * sph_inverse_simple) + p * hgcrt_mod_right_hpl_simple)))))) -> exists M. ((exists pa_b_hpl_power pa_c_hpl_power. ((forall pa_i_hpl_power_repeat. (exists pa_lt_hpl_power_repeat_bound. pa_lt_hpl_power_repeat_bound + S pa_i_hpl_power_repeat = k + j) -> (((exists pa_h_hpl_power_repeat_decoded. pa_h_hpl_power_repeat_decoded + S (p) = S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_repeat_decoded. pa_b_hpl_power = pa_q_hpl_power_repeat_decoded * S ((S (pa_i_hpl_power_repeat)) * pa_c_hpl_power) + (p)))) /\ (exists pa_u_hpl_power_product pa_v_hpl_power_product. ((((exists pa_h_hpl_power_product_start. pa_h_hpl_power_product_start + S (1) = S ((S (0)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_start. pa_u_hpl_power_product = pa_q_hpl_power_product_start * S ((S (0)) * pa_v_hpl_power_product) + (1))) /\ ((((exists pa_h_hpl_power_product_terminal. pa_h_hpl_power_product_terminal + S (M) = S ((S (k + j)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_terminal. pa_u_hpl_power_product = pa_q_hpl_power_product_terminal * S ((S (k + j)) * pa_v_hpl_power_product) + (M))) /\ forall pa_i_hpl_power_product. (exists pa_lt_hpl_power_product_bound. pa_lt_hpl_power_product_bound + S pa_i_hpl_power_product = k + j) -> exists pa_p_hpl_power_product pa_r_hpl_power_product pa_s_hpl_power_product. ((((exists pa_h_hpl_power_product_factor. pa_h_hpl_power_product_factor + S (pa_p_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power)) /\ exists pa_q_hpl_power_product_factor. pa_b_hpl_power = pa_q_hpl_power_product_factor * S ((S (pa_i_hpl_power_product)) * pa_c_hpl_power) + (pa_p_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_partial. pa_h_hpl_power_product_partial + S (pa_r_hpl_power_product) = S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_partial. pa_u_hpl_power_product = pa_q_hpl_power_product_partial * S ((S (pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_r_hpl_power_product))) /\ ((((exists pa_h_hpl_power_product_successor. pa_h_hpl_power_product_successor + S (pa_s_hpl_power_product) = S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product)) /\ exists pa_q_hpl_power_product_successor. pa_u_hpl_power_product = pa_q_hpl_power_product_successor * S ((S (S pa_i_hpl_power_product)) * pa_v_hpl_power_product) + (pa_s_hpl_power_product))) /\ pa_s_hpl_power_product = pa_r_hpl_power_product * pa_p_hpl_power_product)))))))) /\ exists r. ((((exists hpl_gap_lift. hpl_gap_lift + S (r) = (M)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. r + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists sph_positive_lift sph_negative_lift. ((exists ff_u_ph_hpl_sph_lift_positive ff_v_ph_hpl_sph_lift_positive. ((((exists fs_h_ph_hpl_sph_lift_positive_body_start. fs_h_ph_hpl_sph_lift_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_start. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_terminal. fs_h_ph_hpl_sph_lift_positive_body_terminal + S (sph_positive_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_terminal. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_positive) + (sph_positive_lift))) /\ forall ff_i_ph_hpl_sph_lift_positive_body_steps. (exists ph_bound_hpl_sph_lift_positive_body_steps. ph_bound_hpl_sph_lift_positive_body_steps + S ff_i_ph_hpl_sph_lift_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_positive_body_steps ff_previous_ph_hpl_sph_lift_positive_body_steps ff_current_ph_hpl_sph_lift_positive_body_steps. ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient. fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_before. fs_h_ph_hpl_sph_lift_positive_body_steps_before + S (ff_previous_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_before. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_previous_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_after. fs_h_ph_hpl_sph_lift_positive_body_steps_after + S (ff_current_ph_hpl_sph_lift_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_after. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_current_ph_hpl_sph_lift_positive_body_steps))) /\ ff_current_ph_hpl_sph_lift_positive_body_steps = ff_previous_ph_hpl_sph_lift_positive_body_steps * r + ff_coefficient_ph_hpl_sph_lift_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_lift_negative ff_v_ph_hpl_sph_lift_negative. ((((exists fs_h_ph_hpl_sph_lift_negative_body_start. fs_h_ph_hpl_sph_lift_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_start. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_terminal. fs_h_ph_hpl_sph_lift_negative_body_terminal + S (sph_negative_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_terminal. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_negative) + (sph_negative_lift))) /\ forall ff_i_ph_hpl_sph_lift_negative_body_steps. (exists ph_bound_hpl_sph_lift_negative_body_steps. ph_bound_hpl_sph_lift_negative_body_steps + S ff_i_ph_hpl_sph_lift_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_negative_body_steps ff_previous_ph_hpl_sph_lift_negative_body_steps ff_current_ph_hpl_sph_lift_negative_body_steps. ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient. fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_before. fs_h_ph_hpl_sph_lift_negative_body_steps_before + S (ff_previous_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_before. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_previous_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_after. fs_h_ph_hpl_sph_lift_negative_body_steps_after + S (ff_current_ph_hpl_sph_lift_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_after. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_current_ph_hpl_sph_lift_negative_body_steps))) /\ ff_current_ph_hpl_sph_lift_negative_body_steps = ff_previous_ph_hpl_sph_lift_negative_body_steps * r + ff_coefficient_ph_hpl_sph_lift_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. sph_positive_lift + M * hgcrt_mod_left_hpl_lift = sph_negative_lift + M * hgcrt_mod_right_hpl_lift))))))) /\ forall z. (((exists hpl_gap_lift. hpl_gap_lift + S (z) = (M)) /\ ((exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. z + m * hgcrt_mod_left_hpl_lift = a + m * hgcrt_mod_right_hpl_lift) /\ (exists sph_positive_lift sph_negative_lift. ((exists ff_u_ph_hpl_sph_lift_positive ff_v_ph_hpl_sph_lift_positive. ((((exists fs_h_ph_hpl_sph_lift_positive_body_start. fs_h_ph_hpl_sph_lift_positive_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_start. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_positive) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_terminal. fs_h_ph_hpl_sph_lift_positive_body_terminal + S (sph_positive_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_terminal. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_positive) + (sph_positive_lift))) /\ forall ff_i_ph_hpl_sph_lift_positive_body_steps. (exists ph_bound_hpl_sph_lift_positive_body_steps. ph_bound_hpl_sph_lift_positive_body_steps + S ff_i_ph_hpl_sph_lift_positive_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_positive_body_steps ff_previous_ph_hpl_sph_lift_positive_body_steps ff_current_ph_hpl_sph_lift_positive_body_steps. ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient. fs_h_ph_hpl_sph_lift_positive_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient. pb = fs_q_ph_hpl_sph_lift_positive_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * pc) + (ff_coefficient_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_before. fs_h_ph_hpl_sph_lift_positive_body_steps_before + S (ff_previous_ph_hpl_sph_lift_positive_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_before. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_previous_ph_hpl_sph_lift_positive_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_positive_body_steps_after. fs_h_ph_hpl_sph_lift_positive_body_steps_after + S (ff_current_ph_hpl_sph_lift_positive_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive)) /\ exists fs_q_ph_hpl_sph_lift_positive_body_steps_after. ff_u_ph_hpl_sph_lift_positive = fs_q_ph_hpl_sph_lift_positive_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_positive_body_steps)) * ff_v_ph_hpl_sph_lift_positive) + (ff_current_ph_hpl_sph_lift_positive_body_steps))) /\ ff_current_ph_hpl_sph_lift_positive_body_steps = ff_previous_ph_hpl_sph_lift_positive_body_steps * z + ff_coefficient_ph_hpl_sph_lift_positive_body_steps)))))) /\ ((exists ff_u_ph_hpl_sph_lift_negative ff_v_ph_hpl_sph_lift_negative. ((((exists fs_h_ph_hpl_sph_lift_negative_body_start. fs_h_ph_hpl_sph_lift_negative_body_start + S (0) = S ((S (0)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_start. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_start * S ((S (0)) * ff_v_ph_hpl_sph_lift_negative) + (0))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_terminal. fs_h_ph_hpl_sph_lift_negative_body_terminal + S (sph_negative_lift) = S ((S (l)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_terminal. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_terminal * S ((S (l)) * ff_v_ph_hpl_sph_lift_negative) + (sph_negative_lift))) /\ forall ff_i_ph_hpl_sph_lift_negative_body_steps. (exists ph_bound_hpl_sph_lift_negative_body_steps. ph_bound_hpl_sph_lift_negative_body_steps + S ff_i_ph_hpl_sph_lift_negative_body_steps = l) -> exists ff_coefficient_ph_hpl_sph_lift_negative_body_steps ff_previous_ph_hpl_sph_lift_negative_body_steps ff_current_ph_hpl_sph_lift_negative_body_steps. ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient. fs_h_ph_hpl_sph_lift_negative_body_steps_coefficient + S (ff_coefficient_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient. nb = fs_q_ph_hpl_sph_lift_negative_body_steps_coefficient * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * nc) + (ff_coefficient_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_before. fs_h_ph_hpl_sph_lift_negative_body_steps_before + S (ff_previous_ph_hpl_sph_lift_negative_body_steps) = S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_before. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_before * S ((S (ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_previous_ph_hpl_sph_lift_negative_body_steps))) /\ ((((exists fs_h_ph_hpl_sph_lift_negative_body_steps_after. fs_h_ph_hpl_sph_lift_negative_body_steps_after + S (ff_current_ph_hpl_sph_lift_negative_body_steps) = S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative)) /\ exists fs_q_ph_hpl_sph_lift_negative_body_steps_after. ff_u_ph_hpl_sph_lift_negative = fs_q_ph_hpl_sph_lift_negative_body_steps_after * S ((S (S ff_i_ph_hpl_sph_lift_negative_body_steps)) * ff_v_ph_hpl_sph_lift_negative) + (ff_current_ph_hpl_sph_lift_negative_body_steps))) /\ ff_current_ph_hpl_sph_lift_negative_body_steps = ff_previous_ph_hpl_sph_lift_negative_body_steps * z + ff_coefficient_ph_hpl_sph_lift_negative_body_steps)))))) /\ (exists hgcrt_mod_left_hpl_lift hgcrt_mod_right_hpl_lift. sph_positive_lift + M * hgcrt_mod_left_hpl_lift = sph_negative_lift + M * hgcrt_mod_right_hpl_lift))))))) -> z = r))